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```PERIODIC MOTION
301
>eriod as the displacement, and that it differs in phase from
p he latter by , as shown in Fig. 132.
236.  Energy of the Particle.The following do not need 'urther explanation.
U
=.- f* Jo
Fdx
T =   mv*
2 7r
lib   f     c\            t)
 (a2-x2 2x
P2
sin2      (t + to). (VII)
P2
(f+t0). (VI)
(VIII)
Thus the total energy of the particle is constant and equals the maximum values of the potential and kinetic energies. The total energy varies, evidently, directly as the square of the amplitude and inversely as the square of the period.
In Fig. 133, T, U, and V are plotted as ordinates and the time as abscissa, with phase relations which correspond to the curves of Fig. 132.
(I) is the TJ and t Curve.
(II) is the T and t Curve. (lll)is the E and t Curve.
FIG. 133.
237. Average Value of the Potential Energy.  Since U may be considered as a function of either x or t, we will find its average value with respect to both variables. Taking 0 and```